Given right triangles ABC and DCB with rt angles at B and C, triangle
ABC's hypotenuse 20 and triangle DCB's hypotenuse 30. The hypotenuses
intersect at point E, a distance of 10 from BC. Find the length of BC.

From a point P on the circumcircle of the triangle ABC perpendiculars
are dropped to the sides AB, BC, CA. Prove that the line joining the
feet of the perpendiculars bisects the line joining the orthocentre of
triangle ABC and point P.

Prove that in the plane of any triangle ABC, with G the centroid, La,
Lb, and Lc the bisectors of angles A, B, and C, Ga, Gb, and Gc the
reflections of line AG about La, BG about Lb, and CG about Lc, the
three lines Ga, Gb, Gc meet in the symmedian point.

Persons A, B, and C stand at the vertices of an equilateral triangle
with 10 meter sides. At the same moment all of them start moving at 1
m/sec. A always heads toward B, B towards C, and C towards A. Will
they meet? How long will it take?